There is in the physics literature a notion of globally hyperbolic maximal compact (GHMC) AdS spacetime. Mess discovered some remarkable analogies between quasifuchsian hyperbolic manifolds and those GHMC AdS spacetimes [1]}, [2]} which for this reason are now sometimes called “quasifuchsian AdS spacetimes”. In this analogy, the role of the conformal metrics at infinity of a quasifuchsian hyperbolic manifold is played, for quasifuchsian AdS spacetimes, by the “left” and “right” hyperbolic metrics. Considering closed surfaces in quasifuchsian AdS spacetimes is equivalent to considering surfaces invariant under a co-compact surface group action in Question \(W^{thin}_{\mathbb {ADS}^3}\) . In that case, existence results are known. The results can then be stated in terms of the induced metrics or third fundamental forms on the boundary of convex domains in quasifuchsian AdS spacetimes. One can mention in particular:
